Population Growth
Exponential:
Continuous addition of births and
deaths at constant rates (b & d)
Such that r = b - d
dN
 rN
dt
Problem: no explicit prediction is made
Solution: isolate N terms on left, and integrate
Result of the integration:
Nt  N0e
Exponential growth
800
700
N (t)
0
1
2
3
4
5
6
7
8
9
10
log N
1 0 0 4 .6 0 5 1 7 0 1 9
600
1 0 5 .1 2 7 1 1 4 .6 5 5
17019
1 1 0 .5 1 7 0 9 2 4 .7 0 5 1 7 0 1 9
1 1 6 .1 8 3 4 2 4 4 .7 5 5
17019
500
1 2 2 .1 4 0 2 7 6 4 .8 0 5 1 7 0 1 9
1 2 8 .4 0 2 5 4 2 4 .8 5 5 1 7 0 1 9
400
1 3 4 .9 8 5 8 8 1 4 .9 0 5 1 7 0 1 9
1 4 1 .9 0 6 7 5 5 4 .9 5 5 1 7 0 1 9
300
1 4 9 .1 8 2 4 7 5 .0 0 5
17019
1 5 6 .8 3 1 2 1 9 5 .0 5 5 1 7 0 1 9
1 6 4 .8 7 2 1 2 7 5 .1 0 5
17019
200
r=0.05
N(t)
time, t:
rt
100
0
0
10
20
30
Time
40
50
Exponential growth relationships
Exponential growth
800
40
700
35
600
Slope of
Curve on
left
500
N(t)
dN
 rN
dt
30
400
300
200
Nt  N0e
100
10
20
30
40
Time
Slope of this curve
Increases with density
20
15
10
5
rt
0
0
0
0
25
50
100
200
300
400
500
Density
Slope of line = r
600
700
800
Exponential growth, log scale
N (t)
0
1
2
3
4
5
6
7
8
9
10
log N
1 0 0 4 .6 0 5 1 7 0 1 9
1 0 5 .1 2 7 1 1 4 .6 5 5 1 7 0 1 9
1 1 0 .5 1 7 0 9 2 4 .7 0 5 1 7 0 1 9
1 1 6 .1 8 3 4 2 4 4 .7 5 5 1 7 0 1 9
1 2 2 .1 4 0 2 7 6 4 .8 0 5 1 7 0 1 9
1 2 8 .4 0 2 5 4 2 4 .8 5 5 1 7 0 1 9
1 3 4 .9 8 5 8 8 1 4 .9 0 5 1 7 0 1 9
1 4 1 .9 0 6 7 5 5 4 .9 5 5 1 7 0 1 9
1 4 9 .1 8 2 4 7 5 .0 0 5 1 7 0 1 9
1 5 6 .8 3 1 2 1 9 5 .0 5 5 1 7 0 1 9
1 6 4 .8 7 2 1 2 7 5 .1 0 5 1 7 0 1 9
7
6
5
4
ln(N)
time, t:
ln Nt  ln N0  rt
3
Linear increase of log
values with time is a
sign of exponential growth
2
1
0
0
10
20
30
time
40
50
Geometric Growth
Time is measured in discrete (contant) chunks
Simplest approach: Generations are the time unit
R0: Average number of offspring produced per individual,
per lifetime-- Factor that a population will be multiplied by
for each generation. Often called the Net Rate of Increase.
NT  N0 R
T
0
Time is measured in generations in this equation.
Relationship between R0 and r
A population growing for one generation should show the
same result using either of the following equations:
Continuous, where t=t
(t  “generation time”)
Discrete, where T=1
generation
Nt  N0e
NT  N0 R
T
0
rt
Nt  N0 e
rt
N1  N0 R  N 0R0
1
0
If these give the same result, then
rt
N0e  N0 R0
R0 and r
rt
N0e  N0 R0
rt
e  R0
rt  ln R0
r
ln R0
t
So! Information about R and t can lead us to r
Ways of finding R0 and t
R0   lx mx
x
 xl m
t
l m
x
x
x
x
x
x
Cohort study
x
0
1
2
3
4
nx
43
21
10
2
0
total offs pring
44
32
2
0
Survivorship calculations
x
0
1
2
3
4
nx
43
21
10
2
0
total offs pring
44
32
2
0
lx
lx
1
0 .4 8 8 3 7 2 0 9
0 .2 3 2 5 5 8 1 4
0 .0 4 6 5 1 1 6 3
0
1.2
Survivorship, lx
1
0.8
0.6
lx
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Age, x
3
3.5
4
4.5
Fecundity calculations
x
0
1
2
3
4
nx
43
21
10
2
0
total offs pring
44
32
2
0
lx
mx
1
0
0 .4 8 8 3 7 2 0 9 2 .0 9 5 2 3 8 1
0 .2 3 2 5 5 8 1 4
3 .2
0 .0 4 6 5 1 1 6 3
1
0
0
mx
3.5
Fecundity, mx
3
2.5
2
mx
1.5
1
0.5
0
0
1
2
Age, x
3
4
Age-specific reproduction
x
0
1
2
3
4
nx
43
21
10
2
0
total offs pring
lx
mx
lxmx
1
0
0
4 4 0 .4 8 8 3 7 2 0 9 2 .0 9 5 2 3 8 1 1 .0 2 3 2 5 5 8 1
3 2 0 .2 3 2 5 5 8 1 4
3 .2 0 .7 4 4 1 8 6 0 5
2 0 .0 4 6 5 1 1 6 3
1 0 .0 4 6 5 1 1 6 3
0
0
0
0
N et Rate of I nc reas e=
1 .8 1 3 9 5 3 4 9
Offspring per initial individual, lxmx
lxmx
1.2
Generation time
1
0.8
0.6
lxmx
0.4
Ro=area under curve
0.2
0
0
1
2
Age, x
3
4
nx
43
21
10
2
0
Generation Time, t
total offs pring
lx
mx
lxmx
xlxmx
1
0
0
0
4 4 0 .4 8 8 3 7 2 0 9 2 .0 9 5 2 3 8 1 1 .0 2 3 2 5 5 8 1 1 .0 2 3 2 5 5 8 1
3 2 0 .2 3 2 5 5 8 1 4
3 .2 0 .7 4 4 1 8 6 0 5 1 .4 8 8 3 7 2 0 9
2 0 .0 4 6 5 1 1 6 3
1 0 .0 4 6 5 1 1 6 3 0 .1 3 9 5 3 4 8 8
0
0
0
0
0
N et Rate of I nc reas e=
1 .8 1 3 9 5 3 4 9 2 .6 5 1 1 6 2 7 9
G eneration time=
1 .4 6 1 5 3 8 4 6
r~
lxmx
Offspring per initial individual, lxmx
x
0
1
2
3
4
0 .2 1 6 0 1 9 0 9
1.2
Generation time
1
0.8
0.6
lxmx
0.4
Ro=area under curve
0.2
0
0
1
2
Age, x
3
4
Approximate r
x
0
1
2
3
4
nx
43
21
10
2
0
total offs pring
lx
mx
lxmx
xlxmx
1
0
0
0
4 4 0 .4 8 8 3 7 2 0 9 2 .0 9 5 2 3 8 1 1 .0 2 3 2 5 5 8 1 1 .0 2 3 2 5 5 8 1
3 2 0 .2 3 2 5 5 8 1 4
3 .2 0 .7 4 4 1 8 6 0 5 1 .4 8 8 3 7 2 0 9
2 0 .0 4 6 5 1 1 6 3
1 0 .0 4 6 5 1 1 6 3 0 .1 3 9 5 3 4 8 8
0
0
0
0
0
N et Rate of I nc reas e=
1 .8 1 3 9 5 3 4 9 2 .6 5 1 1 6 2 7 9
G eneration time=
1 .4 6 1 5 3 8 4 6
r~
0 .2 1 6 0 1 9 0 9
Assumptions of exponential or
geometric growth projections
Constant lx and mx schedules
This implies that reproduction and survival
will not change with density
This also implies that any changes in physical
or chemical environment have no influence on
survival or reproduction
No important interactions with other species
if age-specific data are used, assume stable age
distribution.
Suppose we let lx, mx and t vary
with density
Bottom line: let r (per capita growth rate) vary with N
r
dN/Ndt
0
0
K
N
Density-dependent growth
r
dN/Ndt
-r/K
0
0
N
Y = A + BX
dN
r
r  N
Ndt
K
K
Logistic equation
dN
N

 rN 1 

dt
K
Predictive form:
Nt 
K
 K  N 0   rt
1
e
 N0 
Human rates of change vs N
0.025
0.02
0.015
y = -0.0019x + 0.0254
0.01
0.005
0
0
2
4
6
8
10
12
14
16
Projection based on Logistic model:
14
12
10
8
6
4
2
0
1950
2000
2050
2100
2150
Earlier US projection, similar
approach:
Logistic Examples
Full-loop (2x the bacteria)
Half-loop (half that on right)
Paramecium, 2 species, growing for 8 days at high <r> and low <l> resource
levels. Scale has been stretched on right to be equivalent to that on the left
More logistic examples
Growth of flour beetles in flower,
In containers holding different amts
of flour
Growth of a zooplankton crustacean, Moina, at different
temperatures
Drosophila studies
Evolution of K in Drosophila
Post-radiation
Hybrid
Control
Inbred
Results suggest that K responds to an increase in genetic variation,
And that it changes gradually through time in response to selection.
Assumptions of Logistic Growth
Constant environment (r and K are constants)
Linear response of per capita growth rate to density
Equal impact of all individuals on resources
Instantaneous adjustment of population growth to change in N
No interactions with species other than those that are food
Constantly renewed supply of food in a constant quantity
Discrete Model for Limited Growth
Same assumptions, except population grows in bursts with each
Generation-- built-in time lag
Models of this sort show the potential influence that a time lag can
have on population change.
 Nt 
Nt 1  N t  rNt 1 

K
Simple model, complex behavior
Discrete Model
1200
1000
N
800
600
400
200
0
0
20
40
60
80
Time (generations)
R = 0.1, K = 1000
100
120
Simple model, complex behavior
Discrete Model
1200
1000
N
800
600
400
200
0
0
20
40
60
80
Time (generations)
R = 1.9, K = 1000
Damped oscillation
100
120
Simple model, complex behavior
Discrete Model
1400
1200
1000
N
800
600
400
200
0
0
20
40
60
Time (generations)
r= 2.2, K = 1000
Limit cycle
80
100
120
Simple model, complex behavior
Discrete Model
1400
1200
1000
N
800
600
400
200
0
0
20
40
60
Time (generations)
r= 2.5, K = 1000
4-point cycle
80
100
120
Simple model, complex behavior
Discrete Model
1400
1200
1000
N
800
600
400
200
0
0
20
40
60
80
Time (generations)
r= 2.58, K = 1000
8-point cycle
100
120
Simple model, complex behavior
Discrete Model
1400
1200
1000
N
800
600
400
200
0
0
20
40
60
Time (generations)
r= 2.7, K = 1000
Erratic
80
100
120
Chaos
Discrete Model
1400
1200
1000
N
800
600
400
200
0
0
20
40
60
Time (generations)
r= 3, K = 1000
80
100
120
Overshoot, Crash, Extinction
Discrete Model
1600
1400
1200
N
1000
800
600
400
200
0
0
20
40
60
80
Time (generations)
r= 3.000072, K = 1000
100
120
Concerns about Chaos
Biological populations don’t appear to have the growth
capacity to generate chaos, but this shows the potential
importance of time lags.
More complicated models can be even more sensitive
Some systems might be completely unpredictable
Evolution of Life Histories
Life history features:
Rates of birth, death, population growth
Patterns of reproduction and mortality
Behavior associated with reproduction
Efficiency of resource use, and carrying capacity
Anything that affects population growth
Patterns
More patterns
Tradeoff